"One popular way of remembering these relationships is by remembering this statement." is grammatically correct. I wouldn't say that the phrase "by remembering this statement" modifies "one popular way of remembering these relationships", though. The verb in the sentence is "is", a linking verb which sets two things equivalent to each other. So the sentence is saying that two things, "a way of remembering the relationships" and "by remembering the statement" are the same.
"Is" can link the subject of a sentence to a word modifying that subject if it is an adjective. Example: "The box is red." You could say that "red" is a modifier of "box". But when the linking verb "is" links two nouns, or things, you don't say that one noun modifies the other. Example: "Bob is my uncle" means "Bob" = "my uncle". You wouldn't say that "my uncle" modifies "Bob" there.
Lastly, I'm a little confused why you decided to reword the sentence as, "One popular way of remembering these relationships by remembering this statement seems to be no effective". Can you explain the origin of the last five words in that sentence? "Seems to be no effective", itself, is not good grammar as written, and it's not crystal clear to me what you want the full sentence to say.
